Measurement method for determining the nonlinearities in an optical fiber

ABSTRACT

A measurement method is provided for determining nonlinearities in an optical fiber wherein, in a first step, at least one optical test signal is injected into the optical fiber, whose test signal power is varied, and a first onset threshold for the stimulated Brillouin scatter is determined on the basis of the change in power of the back scattered optical signal and, in a second step, in addition to the optical test signal, at least one modulated optical pump signal is injected with a predetermined pump signal power and at a first pump wavelength into the optical fiber, and a second onset threshold for the stimulated Brillouin scatter is determined on the basis of the change in the test signal power, and the nonlinearity coefficient of the optical fiber is determined by evaluating at least the first and the second onset threshold, the test and pump signal parameters and the fiber parameters.

BACKGROUND OF THE INVENTION

Nonlinear effects, such as self-phase modulation, cross-phase modulationand four-wave mixing, are known in optical transmission systems; inparticular, in transmission systems which operate on the WDM principle(wavelength division multiplexing). These cause signal distortion in theoptical signal to be transmitted in the optical fiber. Nonlinear effectssuch as these in an optical fiber can be described by the nonlinearitycoefficient.

In order to determine the nonlinearity coefficient of an optical fiber,the publication by Y. Namihira, A. Miyata, N. Tanahashi, “Nonlinearitycoefficient measurements for dispersion shifted fibres using self-phasemodulation method at 1.55 μm”, Electronic Letters, 1994, Vol. 30, No.14, pages 1171-1172, for example, discloses a measurement arrangement inwhich the nonlinearity characteristics of an optical fiber aredetermined by using the self-phase modulation method. Measurementmethods such as these are dependent on access to the start and end ofthe optical fibers to be measured, although this involves considerablemeasurement effort and is virtually impossible in already existingoptical communications networks; that is to say, in optical fibers whichhave already been laid. In addition, a separate return channel isrequired from the fiber end to the fiber start in order to transmit themeasured information.

An object to which the present invention is directed is to improve thedetermination of the nonlinearities in an optical fiber, and to allowthe nonlinearities of an optical fiber to be measured at one end; thatis to say, at the start or at the end of the optical fiber.

SUMMARY OF THE INVENTION

A major aspect of the measurement method according to the presentinvention is that, in a first step, at least one optical test signal isinjected into the optical fiber, whose test signal power is varied, anda first onset threshold for the stimulated Brillouin scatter isdetermined on the basis of the change in power of the backscatteredoptical signal. Furthermore, in a second step, in addition to theoptical test signal, at least one modulated optical pump signal isinjected with a predetermined pump signal power and at a first pumpwavelength into the optical fiber, and a second onset threshold for thestimulated Brillouin scatter is determined on the basis of the change inthe optical test signal power. Finally, the nonlinearity coefficient ofthe optical fiber is determined by evaluation of at least the first andthe second onset threshold, of the test and pump signal parameters, andthe fiber parameters. It is particularly advantageous that themeasurement method according to the present invention makes it possibleto determine the nonlinearity coefficient via a measurement at only oneend; that is to say, at the receiving end or transmitting end. This isan enormous advantage, particularly for the determination of the fibernonlinearities of optical fibers which have already been laid.

In a second embodiment of the measurement method for determining thenonlinearities in an optical fiber, in a first step, at least oneoptical test signal is injected with a test signal power and at a testsignal wavelength into the optical fiber, and the power of thebackscattered optical signal is measured, and a first ratio is formedfrom the injected test signal power and the power of the backscatteredoptical signal. Furthermore, in a second step, in addition to theoptical test signal which has a test signal power and is at a testwavelength, at least one modulated optical pump signal is injected withan adjustable pump signal power and at a first pump wavelength into theoptical fiber, and the power of the backscattered optical signal ismeasured, and a second ratio is determined from the injected test signalpower and the power of the backscattered optical signal. Here, theadjustable pump signal power of the modulated optical pump signal isincreased or decreased until the second ratio matches the first ratio.In this case, the nonlinearity coefficient of the optical fiber is thendetermined by evaluation of the test and pump signal parameters as wellas the fiber parameter. The variation according to the present inventionof the pump signal power of the modulated optical pump signalalternatively makes it possible to determine the nonlinearitycoefficient of the optical fiber by ratio formation, evaluating theobserved fiber parameters and trial parameters.

A further advantage of the measurement method according to the presentinvention is that the test and pump signal parameters which areevaluated on the basis of the first variant of the measurement methodaccording to the present invention are the test signal wavelength, thepredetermined pump signal power, the first pump wavelength and themodulation frequency of the optical pump signal. Furthermore, the testsignal power, the test signal wavelength, the pump signal power that isset, the first pump wavelength, and the modulation frequency of theoptical pump signal are evaluated as the test and pump signalparameters; crucial for the second embodiment of the measurement methodaccording to the present invention.

Theoretical principles relating to the measurement method according tothe present invention for determination of the nonlinearities and thedispersion in an optical fiber will be explained in the following text.

In optical fibers, the nonlinear effect of “stimulated Brillouinscattering (SBS)” occurs as a function of the injected power of anoptical test signal or signal. This narrowband SBS effect with a linewidth of Δν_(B)≈25 MHz, which is governed by the phonon life is known(in this context, see Govind P. Algrawal “Nonlinear Fiber Optics”,Academic Press, 1995, pages 370 to 375). Furthermore, U.S. PatentSpecification U.S. Pat. No. 3,705,992 disclosed the onset threshold forSBS being increased in proportion to the ratio of the spectral widthΔν_(s) of the optical signal which is injected into the optical fiber tothe line width Δν_(B); that is to say,

I _(SBS)˜Δν_(s)/Δν_(B)

where I_(SBS)=intensity of the injected optical signal at the SBS onsetthreshold

In this case, the governing factor for reaching the SBS onset thresholdis the energy which is spectrally integrated in a frequency separationof width Δν_(B). In standard monomode fibers, the SBS onset thresholdoccurs, for example, at slightly below 10 mW for unmodulated opticalsignals or test signals, and at a level which is higher by a factor of 2to 3 dB for binary amplitude-modulated optical signals. The increase forbinary amplitude-modulated optical signals is due to the fact that theoptical signal power is shared between modulation sidebands and thecarrier signal and, particularly at data rates in the Gbit/s range, thepower of the data signal is distributed over a broad spectral band.

In the case of amplitude-modulated signals, SBS leads to signaldistortion due to overmodulation (see, in particular, H. Kawakani,“Overmodulation of Intensity modulated Signals due to stimulatedBrillouin scattering”, Electronic Letters, Volume 30, No. 18, pages 1507to 1508), since the carrier of the amplitude-modulated optical signal,in which the spectral energy density for chip-free modulation isidentical to the laser light source, essentially experiences severeadditional attenuation due to the SBS.

The SBS onset threshold can be increased considerably by considerablyreducing the spectral energy density of the optical signal, integratedover a frequency band of width Δν_(B). Thus, in the case ofamplitude-modulated optical signals, the carrier signal power, measuredwith a resolution of Δν_(B), should be reduced to values which areconsiderably below the SBS threshold power. A reduction such as this canbe achieved by frequency modulation or phase modulation.

The SBS effects in the optical fiber occur essentially within the first20 km (effective length L_(eff)) in a standard monomode fiber. In thiscase, the optical signal requires the following time:$\tau = {\frac{L_{eff} \cdot n}{c}\left( {= {{0.1\quad {ms}\quad {for}\quad L_{eff}} = {20\quad {km}}}} \right)}$

to pass through the effective length L_(eff). In order to reduce SBSeffects, the optical injected power per frequency separation Δν_(B),averaged over a time interval, should be very much less than the time τspent below the SBS threshold power. This requirement makes it possibleto derive the necessary relationship between the modulation shift andthe modulation frequency for various forms of modulation, for SBSsuppression via frequency modulation and amplitude modulation.

In order to reduce the spectrally narrow carrier line of the opticalsignal and to uniformly distribute its power over as many lines, whichare newly created by the phase modulation, as possible, with a frequencyinterval of more than Δν_(B), the phase modulation should be carried outusing modulation frequencies>Δν_(B). As the phase shift increases,${{that}\quad {is}\quad {to}\quad {say}\quad {the}\quad {modulation}\quad {index}\quad m} = \frac{\Delta \quad f_{p}}{f_{m}}$$\begin{matrix}{{{{where}\quad \Delta \quad f_{p}} = {{peak}\quad {frequency}\quad {error}}};} \\{{f_{m} = {{modulation}\quad {frequency}}};}\end{matrix}$

the spectral power per frequency separation decreases. Such amplitudemodulation in the optical fiber can be produced, for example, by thenonlinear effect of cross-phase modulation (XPM) via the additionalinjection of highly amplitude-modulated pump signals in addition to theoptical signals. In this case, the phase modulation produced bycross-phase modulation (XPM) has an RC low-pass filter response alongthe optical fiber. The cut-off frequency ω_(g) of the “low-pass filterresponse” decreases linearly as the channel separation increases, owingthe dispersion-dependent slip in the WDM transmission channels. In orderto achieve effective phase modulation over a broad wavelength band viaXPM, it is necessary to choose the magnitude of the modulation frequencyto be as low as possible, although this should never be below the linewidth Δν_(B).

The intensity I_(SBS) of the backscattered optical signal due to SBS atthe fiber start increases in the backward direction as the injectedoptical signal power increases in accordance with the followingexponential relationship; in this context, see Govind P. Algrawal“Nonlinear Fiber Optics”, 1995, Section 9.2.1: $\begin{matrix}{{I_{SBS}(0)} = {{I_{SBS}(z)}*{\exp \left( {{g_{B}*I_{S}*L_{eff}} - {\alpha*z}} \right)}}} & \left( {A\text{-}1} \right) \\{L_{eff} = {\frac{1}{\alpha^{*}}\left( {1 - {\exp \left( {{- \alpha}*z} \right)}} \right)}} & \left( {A\text{-}2} \right)\end{matrix}$

where

If the amplitude-modulated optical pump signal which produces the XPMand the optical signal propagate simultaneously in the fiber, theoptical signal is increasingly phase-modulated on the basis of the XPMas the path length increases. By way of example, phase modulation with aphase shift of 1.435 rad distributes the spectral power of the carriersignal over a number of frequencies in this case; that is to say, forexample, uniformly between the carrier wave and the two first sidebands.If the modulation frequency is, in this case, greater than the SBS linewidth Δν_(B), then only just ⅓ of the spectral energy density isavailable to form the SBS, for example; that is to say, the SBS onsetthreshold is increased by a factor of 3 from the point at which such aphase shift is reached by the XPM. The local SBS onset threshold thuscan be calculated as a function of the characteristics of the injectedmodulated pump signal and of the optical fiber, as well as of theinjected optical signal, and the SBS onset threshold which results fromthis and is dependent on the optical pump signal can be determined forthe entire fiber.

The SBS onset threshold in the presence of the optical pump signal canbe calculated by breaking down the fiber into small part sections inconjunction with equation (A-1). To a first approximation, the fiber isinitially broken down into n=2 part sections, from which it followsusing equations (A-1) and (A-2) for a fiber of length Z/2 that:

I _(SBS)(z/2)=I _(SBS)(z)*exp(g _(B) *I_(s)*exp(−α*z/2)*1/α*(1−exp(−α*z/2))−αz/2)  (A-3)

and

I _(SBS)(0)=I _(SBS)(z/2)*exp(g_(B) *I_(s)*1/α*(1−exp(−α*z/2))−αz/2)  (A-4)

If the path is broken down into n path sections: $\begin{matrix}{{I_{SBS}(0)} = {{I_{SBS}(z)}*\exp \quad {{}\left\lbrack {g_{B}*I_{S}*\quad {{{\left\{ {1 + {\sum\limits_{k = 1}^{n - 1}{\exp \left( {{- \alpha}*{k/n}*z} \right)}}} \right\}*{1/\alpha}*\left( {1 - {\exp \left( {{- \alpha}*z} \right)}} \right)} - \quad \left. {\alpha \quad z} \right\rbrack}}} \right.}}} & \left( {A\text{-}5} \right)\end{matrix}$

The following text considers the 2nd path section, equation (A-3).Taking account of the spectral change in the optical signal I_(s)resulting from the XPM which is induced by a sinusoidallyamplitude-modulated optical pump signal I_(p), in the fiber, thisresults in addition to the path attenuation exp(−α*z/2) in a furtheradditional attenuation for the carrier by the attenuation factor:

J ₀ ²(m)=J ₀ ²(ξ*γ*I _(p)*1/α*(1−exp(−α*z/2)),  (A-6)

where m is the phase shift or modulation index caused by the XPM on thefirst path section of length z/2, and ξ is a polarization-dependentconstant. For randomly varying polarization: ξ=8/9

Investigations have shown that the carrier of the amplitude-modulatedoptical pump signal (that is to say J₀ ²(m)) is substantially includedin the change in the intensity of the backscattered optical signal. Fromthis, it follows for equation (A-3) with (A-6):

I _(SBS)(z/2)=I _(SBS)(z)*exp[g _(B) *I_(s)*1/α*(1−exp(−α*z/2))*exp(−α*z/2)*J ₀ ²(m(z/2))−αz/2)  (A-7)

where

m(x)=ξ*γ*I _(p)*1/α*(1−exp (−α*x))  (A-8)

Substituting equation (A-7) in equation (A-4) gives the intensity of thebackscattered optical signal I_(SBS), with approximate consideration ofthe XPM.

I _(SBS)(0)=I _(SBS)(z)*exp

[g _(B) *I _(s)*exp(−α*z/2)*J ₀ ²

(m(z/2))*1/α**

(1−exp(−α*z/2))*1/α*(1

−exp(−α*z/2))−αz/2]

**exp[g _(B) *I _(s)*1/α*(1

−exp(−α*z/2))−αz/2)=

=I _(SBS)(z)*exp[g _(B) *I _(s)*1/α*(1

−exp(−α*z/2))**{1

+exp(−α*z/2)*J ₀ ²(m

(z/2)))}−αz]

In order to improve the accuracy, the fiber is broken down into nsubelements (equation A-5)), thus, by an analogous procedure, resultingin: $\begin{matrix}{{I_{SBS}(0)} = {{I_{SBS}(z)}*{\exp \left\lbrack {{g_{B}*I_{S}*{1/\alpha}*\left( {1 - {\exp \left( {{- \alpha}*{z/n}} \right)}} \right)*\left\{ {1 + {\sum\limits_{k = 1}^{n - 1}{{\exp \left( {{- \alpha}*{k/n}*z} \right)}*{J_{0}^{2}\left( {m\left( {k*{z/n}} \right)} \right)}}}} \right\}} - {\alpha \quad z}} \right\rbrack}}} & \left( {A\text{-}9} \right)\end{matrix}$

Comparison of equation (A-9) with equation (A-1) shows that

L _(eff)=1/α*(1−exp (−α*z))

can be replaced by the expression $\begin{matrix}{{L_{eff}\left( {z,\alpha,\gamma,I_{p}} \right)} = {{1/\alpha}*\left( {1 - {\exp \left( {{- \alpha}*{z/n}} \right)}} \right)*\left\{ {1 + {\sum\limits_{k = 1}^{n - 1}{{\exp \left( {{- \alpha}*{k/n}*z} \right)}*{J_{0}^{2}\left( {m\left( {{kz}/n} \right)} \right)}}}} \right\}}} & \left( {A\text{-}10} \right)\end{matrix}$

The effective length L_(eff) is thus, according to equation (A-10) and(A-8), dependent on the nonlinearity coefficient γ of the optical fiber,and on the optical power of the amplitude-modulated optical pump signalI_(p).

Consideration of the Dispersion

If there is a wide frequency separation between the optical pump signaland the injected optical test signal or signals, this results,especially due to the dispersion-dependent slip between the opticalsignal and the optical pump signal, which occurs in a standard monomodefiber (SSMF), in further dependencies between the effective lengthL_(eff) and the fiber dispersion, the frequency separation (wavelengthseparation) of the optical pump signal and optical test signal, and themodulation frequency of the optical pump signal.

From equation (A-10), this results in the following expression forL_(eff)(z,α,γ,I_(p)) for a power section z consisting of n subelements:$\begin{matrix}{{L_{eff}\left( {z,\alpha,\gamma,I_{p},D,{\Delta \quad \lambda},{f\quad {mod}}} \right)} = {{1/\alpha}*\left( {1 - {\exp \left( {{- \alpha}*{z/n}} \right)}} \right)*\left\{ {1 + {\sum\limits_{k = 1}^{n - 1}{{\exp \left( {{- \alpha}*{k/n}*z} \right)}*{J_{0}^{2}\left( {{m\left( {{kz}/n} \right)}*{{Le}\left( {{k*{z/n}},\alpha,D,{\Delta \quad \lambda},{f\quad {mod}}} \right)}} \right)}}}} \right\}}} & \left( {A\text{-}11} \right)\end{matrix}$

where${{Le}\left( {{k*{z/n}},\alpha,D,{\Delta \quad \lambda},{f\quad {mod}}} \right)} \approx \left( \frac{1 + {\exp \left( {{- 2}*L*\alpha} \right)} - {2*{\exp \left( {{- L}*\alpha} \right)}*{\cos \left( {L*\beta*\omega} \right)}}}{\alpha^{2} + {\beta^{2}*\omega^{2}}} \right)^{\frac{1}{2}}$

where:

L=k*z/n,

β=D*Δλ,

ω=2*π*fmod;

m(kz/n)=ξ*γ*I_(p)*Le;

Le(k*z/n,α,D,Δλ,fmod) describes the variation of the modulation index{m(kz/n)} interalia also as a function of the modulation frequency andof the wavelength separation between the optical pump signal and thetest signal.

For high dispersion values D, high modulation frequencies fmod and widewavelength separation Δλ, L_(eff) once again assumes its original formfrom equation (A-2); that is to say, L_(eff) is dependent only on thefiber attenuation α and the location z, and the SBS suppression causedby the optical pump signal is reduced.

The variation in the SBS onset threshold as a function of the pump,signal and fiber parameters is obtained by substitution of equation(A-10) or equation (A-11) in:

 P _(SBS)=21*A _(eff) /g _(B) /L _(eff)  (A-12)

from Godvind P. Agrawal, “Nonlinear Fiber Optics”, Academic Press, 1995,formula (9.2.6).

The dispersion D and the nonlinearity coefficient y can be determinedfrom equation (A-11) and equation (A-12) from the variation in theeffective length L_(eff)(z,α,I_(p),D,Δλ,fmod) as a function of theoptical pump power I_(p), of the wavelength difference between the pumpsignal and the test signal Δλ, as well as the modulation frequency fmodand the SBS onset threshold P_(SBS).

Based on the measurement method according to the present invention, thenonlinearity coefficient γ and the dispersion D are determined using theshift in the SBS onset threshold P_(SBS) resulting from the change inthe spectrum of the injected optical test signal, which is caused by thecross-phase modulation (XPM) resulting from the sinusoidallyamplitude-modulated optical pump signal in the optical fiber, usingequation (A-12).

In this case, the Brillouin gain constant g_(B) and the effective areaA_(eff) are optical fiber constants, which are naturally available forthe optical fiber to be measured, or can be determined without anysignificant technical effort. However, as already mentioned, theeffective length L_(eff)(z,α,I_(p),D,Δλ,fmod) can be influenced by thetrial conditions and, based on formula (A-11), depends on the length ofthe fiber z, on the fiber attenuation α, on the wavelength differencebetween the optical pump signal and test signal Δλ and the modulationfrequency fmod of the amplitude-modulated optical pump signal.

In the measurement method according to the present invention fordetermination of the nonlinearity coefficient γ of an optical fiber, ifthe wavelength difference between the optical pump signal and the testsignal Δλ is, for example, less than 1 nm, and the modulation frequencyfmod of the amplitude-modulated optical pump signal is less than 200MHz, the effect of the dispersion influence on the measurement resultcan be ignored; that is to say, the effective length L_(eff) does notdepend, to a first approximation, on the fiber dispersion D. Accordingto the present invention, a first SBS onset threshold P_(SBS1) and asecond SBS onset threshold P_(SBS2), which is shifted owing to thecross-phase modulation (XPM) caused by the injected modulated pumpsignal in the optical fiber, are measured, and these are evaluatedtogether with the test and pump signal parameters as well as the fiberparameters using equations (A-11) and (A-12), with the dispersion Dbeing negligible. Alternatively, according to the present invention, afirst measurement and a second measurement of the backscattered opticalpower can be carried out, with only the optical test signal beinginjected into the optical fiber for the first measurement, with apredetermined power and at a predetermined wavelength, and with themodulated optical pump signal in addition to the optical test signalbeing injected into the optical fiber for the second measurement, inorder to produce the cross-phase modulation (XPM). In this case, in boththe first measurement and the second measurement, the injected power ofthe optical test signal is increased until a predetermined ratio isobtained between the injected power of the optical test signal and thebackscattered power. The optical test signal and optical pump signalpowers used for the first and second measurements in the measurementmethod, together with the test and pump signal parameters as well as thefiber parameters, are once again evaluated using equations (A-11) and(A-12), with the dispersion D being negligible.

If, in addition to determination of the nonlinearity coefficient γ, itis also intended to determine the dispersion constant D of an opticalfiber using the measurement method according to the present invention,then the wavelength difference between the optical pump signal and thetest signal Δλ is chosen, by way of example, to be greater than 10 nm;that is to say, the effective length L_(eff) depends on the fiberdispersion D and the wavelength difference between the optical pumpsignal and test signal Δλ. Based on the measurement method according tothe present invention, in addition to the first SBS onset thresholdP_(SBS1), without any injected modulated optical pump signal, a thirdSBS onset threshold P_(SBS3) is determined which has a different profilefrom the second SBS onset threshold P_(SBS2) owing to the change to thepump signal parameters, or a third measurement is carried out in which,in addition to the optical test signal, the changed modulated opticalpump signal is injected into the optical fiber in order to produce thecross-phase modulation (XPM) is used to increase the injected power inthe optical test signal until a predetermined ratio is reached betweenthe injected power in the optical test signal and the backscatteredpower. The detailed procedure of the measurement method according to thepresent invention and the determination of the nonlinearity constant γand of the dispersion constant D will be explained in more detail withreference to the following exemplary embodiment.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 shows, by way of example, a measurement arrangement for carryingout the measurement method according to the present invention.

FIG. 2 shows, in a diagram, the shift according to the present inventionin the SBS onset threshold.

FIG. 3 shows, in a further calculated diagram, the measurement methodaccording to the present invention for determining of the nonlinearityconstants and dispersion constants.

DETAILED DESCRIPTION OF THE INVENTION

By way of example, FIG. 1 uses a block diagram to illustrate ameasurement arrangement MAO for carrying out the measurement methodaccording to the present invention for determination of thenonlinearities in an optical fiber OF in which, by way of example, anoptical standard monomode fiber OF has been chosen as the test object inFIG. 1. The measurement arrangement MAO illustrated in FIG. 1 has a testsignal unit TSU, a pump signal unit PSU, an optical coupler OK, acontrollable switching unit S, a circulator Z, a filter unit FU, ameasurement transducer MW and a control and evaluation unit CU. The testsignal unit TSU has a control input ri and a signal output e, which isconnected to the circulator Z via the optical coupler OK and via thefirst optical connecting line VL1. The optical coupler OK is, in turn,connected via a second optical connecting line VL2 to the output e ofthe controllable switching unit S. The circulator Z is also connectedvia a third optical connecting line VL3 to the test object (the opticalfiber OF) and via a first supply line ZL1 to the input fi of the filterunit FU, whose output fe is connected via a second supply line ZL2 tothe input i of the measurement transducer MW. The output e of themeasurement transducer MW is connected via an electrical supply line EZLto the control and evaluation unit CU, which is connected via a controlline SL to the control input si of the controllable switching unit S,via a first control line RL1 to the control input ri of the test signalunit TSU, and via a second control line RL2 to the control input ri ofthe pump signal unit PSU. The pump signal unit PSU furthermore has afirst and a second output e1, e2, which are connected to the first andsecond input i1, i2 of the controllable switching unit S. An opticalcoupler OK also may be used, for example, instead of the circulator Z(not illustrated in FIG. 1).

The control and evaluation unit CU contains a first power regulator RL1,a second power regulator LR2, a switch-mode regulator SR, a memory unitMEM, an evaluation unit AE and a control unit MC (for example,incorporated in a microprocessor). The memory unit MEM, the first andsecond power regulators RL1, RL2, the switch-mode regulator SR and theevaluation unit AE are connected to the control unit MC, with the firstand second power regulators LR1, LR2 also being connected to theevaluation unit AE. Furthermore, the first power regulator LR1 isconnected via the first control line RL1 to the test signal unit TSU,the second power regulator LR2 is connected via the second control lineRL2 to the pump signal unit PSU, and the switch-mode regulator SR isconnected via the switching line SL to the controllable switching unitS, while the evaluation unit AE is connected via the electrical supplyline EZL to the measurement transducer MW.

The measurement method according to the present invention is carriedout, for example, on the basis of a measurement routine MR and anassessment routine BWR in the control unit MC, which, inter alia,regulate or control the test signal unit TSU, the pump signal unit PSUand the controllable switching unit S. An optical test signal ots isthus produced in the test signal unit TSU, at a predetermined firstwavelength λ1 and with a predetermined power P_(s), in which case theoptical test signal ots also may be amplitude-modulated, for example,with a first modulation frequency ν1. According to the presentinvention, the optical test signal ots is thus injected in modulatedform or in unmodulated form into the test object; that is to say, intothe optical fiber OF. In the measurement arrangement MAO illustrated inFIG. 1, the optical test signal ots is, for example, transmitted via theoptical coupler OK and via the first distributor line VL1 to thecirculator Z, and is injected into the optical fiber OF from thecirculator Z via the third distributor line VL3. For the first step ofthe method according to the present invention, only the optical testsignal ots is injected into the optical fiber OF; that is to say, thecontrollable switching unit S does not pass on to the optical coupler OKany of the optical pump signals ops which are produced in the pumpsignal unit PSU. As such, the controllable switching unit S uses acontrol command ss which is produced in the control regulator SR toconnect the third, unused input i3 of the controllable switching moduleS to the output e.

The nonlinear effect of “stimulated Brillouin scattering (SBS)” isformed in the optical fiber OF as a function of the injected test signalpower P_(s) of the optical test signal ots. This narrowband nonlinearSBS effect results in a portion of the optical test signal ots beingscattered or reflected back in the opposite direction to the injectiondirection. This backscattered optical signal ros is passed via thecirculator Z and via the first supply line ZL1 to the filter input fi ofthe filter unit FU. In the filter unit PU, such as a bandpass filterwith a narrow pass band around the first wavelength λ1 of the opticaltest signal ots, the backscattered optical signal ros is filtered, andthe filtered backscattered signal gros is emitted at the filter outputfe. The filtered backscattered signal gros is then transmitted via thesecond supply line ZL2 to the input i of the measurement transducer MW,such as an optoelectrical transducer, where the measurement transducerMW converts it to an electrical signal es. The electrical signal es issupplied to the control and evaluation unit CU and/or to the evaluationunit AE via the electrical supply line EZL, in which the power P_(ros)of the electrical signal es, and hence of the backscattered opticalsignal ros, is determined and/or assessed.

The evaluation unit AE, which is controlled by the control unit MC,determines the power P_(ros) of the backscattered optical signal ros,and an assessment routine BWR is used to compare the determinedbackscattered signal power P_(ros) with the power P_(s), which is storedin the memory unit MEM, of the optical test signal ots. The first powerregulator LR1, controlled by the measurement and assessment routine MR,BWR in the control unit MC, uses the comparison result to form a firstcontrol signal rs1 in order to increase or, if appropriate, to reducethe power P_(s) of the optical test signal ots. In consequence, thepower P_(s) of the optical test signal ots is, for example, increaseduntil a first onset threshold SBS₁ for stimulated Brillouin scatter isreached; that is to say, the power P_(ros) of the backscattered signalros corresponds, for example, to {fraction (1/10)} of the power P_(s) ofthe injected test signal ots. The value of the first critical power,which is emitted at the time when the first onset threshold SBS₁ ofstimulated Brillouin scatter is reached, and/or the first SBS onsetthreshold P_(s1) for the optical test signal ots are/is stored in thememory unit MEM in accordance with the measurement routine MR.

According to the present invention, in a second step of the measurementmethod, in addition to the modulated or unmodulated optical test signalots, at least one modulated optical pump signal ops is injected into theoptical fiber OF with a predetermined first pump signal power P_(P1) andat a first wavelength λ1. For this purpose, an optical pump signal opsis amplitude-modulated with a first wavelength λ1 and, in addition, theoptical pump signal ops is amplitude-modulated with a first modulationfrequency ν1, in the optical pump signal unit PSU, in which case theamplitude modulation may, for example, be in the form of sinusoidal,square-wave or sawtooth-waveform amplitude modulation.

The modulated optical pump signal ops is emitted at the first output e1of the pump signal unit PSU to the first input i1 of the controllableswitching unit S. Based on the second step of the measurement methodaccording to the present invention, the measurement routine MR (which iscarried out in the control unit MC) in the switch-mode regulator SRgenerates a control signal ss in order to connect the first input i1 ofthe controllable switching unit S to the output e, and this controlsignal is transmitted via the control line SL to the controllableswitching unit S. Once the optical pump signal ops has been connectedfrom the first input i1 to the output e of the controllable switchingunit S, the optical pump signal ops is passed via the second distributorline VL2 to the optical coupler OK. The optical coupler OK is used toinject the optical pump signal ops into the first distributor line VL1,and to transmit it, in addition to the optical test signal ots, at thecirculator Z. The circulator Z injects the optical test signal ots andthe optical pump signal ops into the optical fiber OF via the thirdoptical distributor line VL3.

The additional injection of the modulated optical pump signal opsproduces the nonlinear effect of cross-phase modulation (XPM) in theoptical fiber OF and, hence, causes phase modulation of the optical testsignal ots, which broadens the frequency spectrum of the optical testsignal ots. The broadening of the frequency spectrum of the optical testsignal ots first of all results in a decrease in the power of thebackscattered optical signal ros; that is to say, the portion of theinjected optical test signal ots which is scattered or reflected back inthe opposite direction to the injection direction as a result of thenarrowband nonlinear SBS effect decreases. The backscattered opticalsignal ros is, in turn, passed via the circulator Z and via the firstsupply line ZL1 to the filter input fi of the filter unit FU. In thefilter unit FU, the backscattered optical signal ros is filtered, andthe filtered backscattered signal gros is emitted at the filter outputfe. The filtered backscattered signal gros is then once againtransmitted via the second supply line ZL2 to the input i of themeasurement transducer MW, and the measurement transducer MW converts itto an electrical signal es. The electrical signal es is supplied to thecontrol and evaluation unit CU and/or to the evaluation unit AE via theelectrical supply line EZL, in which the power P_(ros) of the electricalsignal es, and hence of the backscattered optical signal ros, aredetermined and/or assessed.

As already described, the evaluation unit AE, which is controlled by thecontrol unit MC, determines the power P_(ros) of the backscatteredoptical signal ros, and the determined backscattered signal powerP_(ros) is compared by the assessment routine BWR with the power P_(s),which is stored in the memory unit MEM, of the optical test signal ots.The first power regulator LR1, controlled by the measurement andassessment routine MR, BWR in the control unit MC, uses the comparisonresult to form the first control signal rs1 in order to increase thepower P_(s) of the optical test, signal ots. The power P_(s) of theoptical test signal ots is increased until a second onset threshold SBS₂for stimulated Brillouin scatter is reached, which is higher than thefirst onset threshold SBS₁; that is to say, the power P_(ros) of thebackscattered signal ros once again corresponds, for example, to{fraction (1/10)} of the power P_(s) of the injected test signal ots.The value of the second critical power P_(s2) and/or of the second SBSonset threshold P_(SBS2) of the optical test signal ots, which isemitted at the time when the second onset threshold SBS₂ for stimulatedBrillouin scatter is reached, is stored in the memory unit MEM inaccordance with the measurement routine MR. Furthermore, the firstoptical pump signal power P_(P1) that is set at that time is stored inthe memory unit MEM.

FIG. 2 uses, by way of example, a diagram to illustrate the first SBSonset threshold SBS₁ and the shifted or raised second SBS onsetthreshold SBS₂. The diagram has a horizontal axis (abscissa) and avertical axis (ordinate), with the power P_(s) of the injected opticaltest signal ots being plotted along the horizontal axis, and the powerP_(ros) of the backscattered optical signal ros being plotted along thevertical axis, in each case in dBm. Based on the first step of themeasurement method according to the present invention, the illustratedfirst SBS onset threshold SBS₁ was measured by injecting the opticaltest signal ots into the optical fiber OF and by raising the test signalpower P_(s) increasingly, with the change or increase in the powerP_(ros) of the backscattered optical signal ros being recorded. Theonset of the nonlinear SBS effect can be seen clearly in the diagramshown in FIG. 2 and, by way of example in the illustrated case, occursat a test signal power P_(s) of about 0.002 watts. Beyond this criticaltest signal power P_(s), it is possible to see a considerably fasterrise in the measurement curve for the first measurement method step, anda considerable rise in the power P_(ros) of the backscattered opticalsignal ros as a result of the SBS. This steep rise of the first SBSonset threshold SBS₁ occurs in a test signal power P_(s) band of about 2dBm and then flattens out once again, so that the profile of the powerP_(ros) of the backscattered optical signal ros with respect to the testsignal power P_(s) assumes approximately the same gradient asimmediately before the first SBS onset threshold SBS₁. Based on thesecond step of the measurement method according to the presentinvention, an optical pump signal ops is injected into the optical fiberOF in addition to the optical test signal ots, so that the cross-phasemodulation XPM which occurs in the optical fiber OF results in the SBSonset threshold being shifted to the right; that is to say, thenonlinear SBS effect occurs at a higher injected test signal powerP_(s). By way of example, for the measurement curve which is illustratedin the diagram and contains the second SBS onset threshold SBS₂, anoptical pump signal ops was injected into the optical fiber OF in such away that amplitude modulation was carried out with a modulationfrequency of 20 MHz, and the pump signal power was 0.2 watts.Furthermore, the wavelength difference Δλ between the optical testsignal ots and the optical pump signal ops was approximately 10 nm. Theapproximately 2 dBm shift in the SBS onset threshold which can be seenfrom the diagram is evaluated together with the known test and pumpsignal parameters as well as the known fiber parameters in order todetermine the nonlinearity coefficient γ. A shift of 1 to 3 dB in theSBS onset threshold is required, by way of example, for uniqueevaluation of the SBS onset thresholds for the purpose of determinationof the nonlinearity coefficient γ according to the present invention.

As has already been indicated in the part of the description whichcovers the theoretical principles for understanding of the presentinvention, the increase in the SBS onset threshold from SBS₁ to SBS₂,which was 2 dB in the example, is represented, by way of example, bycombination of the equations (A-11, A-12) and by forming the ratio ofthe measurement values of the two measurement curves which areillustrated in FIG. 2, as a function of the product of thepolarization-dependent constant ξ, of the nonlinearity constant γ and ofthe injected pump power P_(P1), P_(P2), and the product of thedispersion constant D, the wavelength difference Δλ and the modulationfrequency fmod. Such an evaluation of the measurement curves illustratedin FIG. 2 is illustrated by way of example in the form of a diagram inFIG. 3, in particular with a third measurement curve, which is notillustrated in FIG. 2, being evaluated in FIG. 3. The diagram shows afirst, a second and a third measurement curve MK1, MK2, MK3, which aredetermined from the known measurement parameters determined according tothe present invention. For this purpose, the diagram has a horizontalaxis (abscissa) and a vertical axis (ordinate), with the product of thepolarization-dependent constant ξ, the nonlinearity constant γ and therespectively injected pump power P_(P1), P_(P2) ξ*γ*P_(p), being plottedon a logarithmic scale along the horizontal axis, and with the productof the dispersion constant D, the wavelength difference Δλ and themodulation frequency fmod D*Δλ*fmod being plotted along the verticalaxis. The illustrated measurement curves MK1, MK2, MK3 are obtained fora 100 km long optical fiber OF with an attenuation constant of 0.2 dB,with the product ξ*γ* P_(p) which is plotted on the abscissa having avalue range for the pump power P_(p) of about 0.1 to 2 watts, and theproduct D*Δλ*fmod which is plotted on the ordinate having a value rangefor the wavelength separation Δλ around 10 nm, for a modulationfrequency of 0 to 1 GHz. The first measurement curve MK1 represents anincrease of 1 dB in the first SBS onset threshold SBS₁, the secondmeasurement curve represents an increase of 2 dB, and the thirdmeasurement curve represents an increase of 3 dB, with the pump signalpower P_(p) injected into the optical fiber accordingly being increasedfrom 0.1 watts to 0.2 watts in each for this purpose. Furthermore, afirst, a second, a third and a fourth measurement point MP1 to MP4 aremarked along the second and third measurement curves MK2, MK3 in FIG. 3,and these are the measurement points which are selected, by way ofexample, for determination of the nonlinearity constant γ and of thedispersion constant D using an iterative evaluation method. For themeasurement method according to the present invention for determinationof the nonlinearities in the optical fiber OF, it is sufficient initself to determine at least two measurement values. In the illustratedexemplary embodiment, however, a more comprehensive representation ofthe method according to the present invention is preferred.

In order to determine the nonlinearity coefficient γ, the measurementvalues which are stored in the memory unit MEM and include the first andsecond test signal powers P_(S1), P_(S2), the first pump signal powerP_(P1) as well as the measurement curves illustrated in FIG. 2 and FIG.3 are evaluated using the assessment routine BWR which is carried out inthe control unit MC. With a wavelength difference Δλ of around 1 nmbetween the optical test signal ots and the optical pump signal ops, andwith a low modulation frequency fmod around 200 MHz, the dispersioninfluence on the measurement result is negligible, so that, according toequation (A-12), the effective length L_(eff) does not, to a firstapproximation, depend on the dispersion constant D, and it is thuspossible to determine the nonlinearity constant γ, using the assessmentroutine BWR, by evaluation of equation (A-12).

The second measurement curve MK2 illustrated in FIG. 3, in particularthe first measurement point MP1, will be used by way of example toexplain the process for determination of the nonlinearity constant γusing the assessment routine. The first measurement point MP1 denotesthe intersection of the second measurement curve MK2 with the abscissain FIG. 3, which thus takes account of the negligible dispersionconstant D and of the small wavelength difference Δλ for the situationunder consideration. A negligible ordinate value and a logarithmicabscissa value (10*log10) of −40.9 l/m/W thus can be read as thecoordinates of the first measurement point MP1 from the diagram in FIG.3. The first pump signal power P_(P1), which is stored in the memoryunit MEM is in this case 20 dBm, which corresponds to a first pumpsignal power P_(P1) of 100 mW. Thus, taking account of thepolarization-dependent constant ξ=1, this results in a nonlinearityconstant γ of 0.000813 l/mW after the following conversions:

10*log10(γ*P _(P1))=−40.9 l/m/W

γ*P_(P1)=8.13*10⁻⁵ l/m/W

γ=0.000813 l/m/W

The nonlinearity constant γ can be determined in an analogous manner,using the assessment routine BWR, by way of example at the intersectionof the third measurement curve MK3 with the abscissa of the diagramillustrated in FIG. 3.

A process for determining the dispersion characteristics, that is to saythe dispersion constant D, of the optical fiber OF will be carried outin the following text according to the present invention, such that, ina third step, the amplitude-modulated optical pump signal ops isinjected, in addition to the optical test signal ots, into the opticalfiber OF with the first pump signal power P_(P1) and at a second pumpwavelength λ2 and, a third shifted onset threshold SBS₃ for stimulatedBrillouin scatter is once again determined by changing the power of thebackscattered optical signal ros, by increasing the first pump signalpower P_(P1) until the power P_(ros) of the backscattered signal rosonce again corresponds, for example, to {fraction (1/10)} of the powerP_(s) of the injected test signal ots. As such, if the wavelengthdifference Δλ between the optical test signal ots and the second opticalpump signal ops2 is increased, by way of example, from 1 to 10 nm in thethird step, then the first pump power P_(P1), must be increased by 3 dBin order once again to obtain the second SBS onset threshold SBS₂. Thisresults in a second optical pump signal power P_(P2) for increasing thewavelength difference Δλ in order to reach the second SBS onsetthreshold SBS₂ or, in other words: the increased wavelength differenceΔλ between the optical test signal ots and the second optical pumpsignal ops2 means that the dispersion affects the measurement result insuch a way that the first optical pump signal power P_(P1) must beincreased in order to reach the second SBS onset threshold SBS₂.

This technical effect is evaluated as follows, according to the presentinvention, in order to determine the dispersion constant D. Assuming apolarization-dependent constant ξ=1, the measurement curves illustratedin FIG. 3, in particular the first and third measurement curves MK1,MK3, are used for the determination process via the assessment routingBWR. A first measured pump signal power P_(P1) of 20 dBm was evaluatedin order to calculate the first measurement curve MK1, and a thirdmeasured pump signal power P_(P3) of 26 dBm was evaluated in order tocalculate the third measurement curve MK3, corresponding to an increasein the pump power P_(p) of 6 dB in order to compensate for the increasein the wavelength difference Δλ, that is to say, with a first opticalpump signal ops1 at a first pump wavelength Δλ, by way of example, afirst pump signal power P_(P1) is required in order to reach the secondSBS onset threshold SBS₂, and a second pump signal power P_(P2), whichis greater by a factor of 6 dB, is required when using a second opticalpump signal ops2 at a higher, second pump wavelength λ₂. The thirdmeasurement curve MK3 which is obtained in this way and is shifted tothe right in the diagram in comparison to the first measurement curveMK1 is evaluated starting at the second measurement point MP2 in orderto determine the dispersion constant D using an iterative evaluationmethod. For this purpose, the data record which represents the thirdmeasurement curve MK3 and which is stored in the memory unit MEM isevaluated using the assessment routine BWR such that the intersectionbetween the abscissa and the third measurement curve MK3 is first of allselected as the second measurement point TP2 and, starting from theabscissa value of the second measurement point MP2, the abscissa valueof the fourth measurement point is determined from the data record byshifting to the right, or reducing, the abscissa value of the secondmeasurement point MP2 by the magnitude of the increase in the pumpsignal power P_(P), that is to say, 6 dB in the exemplary embodimentunder consideration. The associated ordinate value of the fourthmeasurement point MP4 is determined from this.

Based on the difference (as considered in the exemplary embodiment)between the first product of the dispersion constant D, the firstwavelength difference Δλ1 and the modulation frequency fmod D*Δλ1*fmodfor the first measurement curve MK1 and the second product of thedispersion constant D, the second wavelength difference Δλ2 and themodulation frequency fmod D*Δλ2*fmod for the third measurement curve MK3of 10, the first measurement point MP1, or starting point, for theiterative assessment method was not sufficiently precise, and isimproved as follows. The previously determined ordinate value of thefourth measurement point MP4 is divided by the difference between thefirst and the second product, the factor 10, and a new improved ordinatevalue is thus determined for the first measurement point MP1. Thedatabase is used to determine the associated new, improved abscissavalue for the first measurement point MK1 with respect to the new,improved ordinate value, and is stored in the memory unit MEM forfurther processing. Based on the first iteration step, the new, improvedabscissa value for the second measurement point MP2 is in turn shiftedto the right, or reduced, by the magnitude of the increase in the pumpsignal power P_(P), that is to say by 6 dB in the exemplary embodimentunder consideration, in a second run of the iterative assessment method.An improved ordinate value for the new fourth measurement point MP4which results in this case, is determined from this. In the majority ofapplications, this assessment method converges after a small number ofiterations, so that the ordinate value which is obtained for the fourthmeasurement point MP4 can be used for determination of the dispersionconstant in accordance with the following equation:

D=4.4*10⁻⁴/(Δλ*fmod)=

=4.4*10⁻⁴/(10⁻⁸*2*10⁸)ps/nm/km =

D=220 ps/nm/km

A dispersion constant of D=220 ps/nm/km is thus obtained for thedescribed exemplary embodiment. The second measurement curve may beevaluated in an analogous manner, by way of example, in order todetermine the associated dispersion constant D.

The measurement arrangement according to the present invention is in noway restricted to a transmission-end implementation, but may be used forany desired optical transmission media, at the receiving end as well.

Indeed, although the present invention has been described with referenceto specific embodiments, those of skill in the art will recognize thatchanges may be made thereto without departing from the spirit and scopeof the present invention as set forth in the hereafter appended claims.

What is claimed is:
 1. A measurement method for determiningnonlinearities in an optical fiber, the method comprising the steps of:injecting, in a first step, at least one optical test signal (ots)having a varying test signal power (P_(s)) into the optical fiber;determining a first onset threshold (SBS₁) for stimulated Brillouinscatter based on a change in power (P_(ros)) of a backscattered opticalsignal (ros); injecting, in a second step, in addition to the opticaltest signal (ots), at least one modulated optical pump signal (ops) witha predetermined pump signal power (P_(P1)) and at a first pumpwavelength (λ₁) into the optical fiber; determining a second onsetthreshold (SBS₂) for the stimulated Brillouin scatter based on a changein the test signal power (P_(s)); and determining a nonlinearitycoefficient (γ) of the optical fiber by evaluating at least the firstand the second onset thresholds (SBS₁, SBS₂), test and pump signalparameters and fiber parameters.
 2. A measurement method for determiningnonlinearities in an optical fiber as claimed in claim 1, wherein a testsignal wavelength (λ₁), the predetermined pump signal power (P_(P1)),the first pump wavelength (λ₁) and a modulation frequency (υ₁) of theoptical pump signal (ops) are evaluated as the test and pump signalparameters.
 3. A measurement method for determining nonlinearities asclaimed in claim 2, wherein the test signal wavelength (λ₁) and thefirst pump wavelength (λ₁) have a wavelength difference of less than 1nm.
 4. A measurement method for determining nonlinearities in an opticalfiber as claimed in claim 2, the method further comprising the steps of:injecting, in a third step, in addition to the optical test signal(ots), the modulated optical pump signal (ops) with the predeterminedpump signal power (P_(P1)) and at a second pump wavelength (λ₂) into theoptical fiber; determining a third shifted onset threshold (MK3) of thestimulated Brillouin scatter again by a change in the power (P_(ros)) ofthe backscattered optical signal (ros); and determining a dispersionconstant (D) of the optical fiber by additionally evaluating at leastthe second and the third onset thresholds (SPS₁, MK3), the testwavelength (λ₁), the pump signal power (P_(P1)), the first and thesecond pump wavelengths (λ₁), (λ₂), the modulation frequency (υ₁) of theoptical pump signal (ops) and the fiber parameters.
 5. A measurementmethod for determining nonlinearities in an optical fiber as claimed inclaim 4, wherein the first, second and third onset thresholds (SBS₁,SBS₂, MK3) of the stimulated Brillouin scatter are each determined viathe test signal power (P_(s)) which causes the onset of the stimulatedBrillouin effect.
 6. A measurement method for determining nonlinearitiesin an optical fiber as claimed in claim 4, wherein the modulationfrequency (υ₁) of the optical pump signal (ops) is chosen to be higherthan an SBS line width (Δυ_(B)).
 7. A measurement method for determiningnonlinearity in an optical fiber as claimed in claim 1, wherein thefiber parameters that are taken into account in the evaluation are aneffective fiber length (L_(eff)), an attenuation constant (α), apolarization factor (ξ) for randomly varying polarization and aBrillouin gain factor (g_(B)) of the optical fiber.
 8. A measurementmethod for determining nonlinearities in an optical fiber as claimed inclaim 1, wherein the optical pump signal (ops) is modulated via at leastone of sinusoidal, square-wave and sawtooth-waveform amplitudemodulation.
 9. A measurement method for determining nonlinearities in anoptical fiber as claimed in claim 1, wherein the evaluation is carriedout in accordance with the formula:P_(s)^(cr) = 21 * A_(eff)/g_(B)/L_(eff)  and${L_{eff}\left( {z,\alpha,\gamma,I_{p},D,{\Delta \quad \lambda},\quad {f\quad {mod}}} \right)} = {{1/\alpha}*\left( {1 - {\exp \left( {{- \alpha}*{z/n}} \right)}} \right)*\quad \left\{ {1 + \quad {\sum\limits_{k = 1}^{n - 1}{{\exp \left( {{- \alpha}*{k/n}*z} \right)}*{J_{0}^{2}\left( {{m\left( {{kz}/n} \right)}*{{Le}\left( {{k*{z/n}},\alpha,D,{\Delta \quad \lambda},{f\quad {mod}}} \right)}} \right)}}}} \right\}}$

where$\left( \frac{1 + {\exp \left( {{- 2}*L*\alpha} \right)} - {2*{\exp \left( {{- L}*\alpha} \right)}*{\cos \left( {L*\beta*\omega} \right)}}}{\alpha^{2} + {\beta^{2}*\omega^{2}}} \right)^{\frac{1}{2}}$

where: L=k*z/n, β=D*Δλ, ω=2*π*fmod; m(kz/n)=ξ*γ*I _(p) *Le; and where:P_(s) ^(cr)=backscattered power at the onset threshold of SBS, g_(B)=aBrillouin gain constant, A_(eff)=an effective area, L_(eff)=an effectivelength, Z=a location variable, α=a fiber attenuation constant, D=adispersion constant, Δλ=a wavelength difference between the test signaland the pump signal, fmod=a modulation frequency of the pump signal,I_(p)=a pump power of the injected pump signal, n=a number ofsubelements for the approximation, γ=a nonlinearity coefficient, ξ=apolarization-dependent constant.